arrangement -- create a hyperplane arrangement

Synopsis

Usage:
arrangement(L,R) or arrangement(M) or arrangement(M,R)
Inputs:
- L, a list of linear equations in the ring R
- R, a polynomial ring or linear quotient of a polynomial ring
- M, a matrix whose rows represent linear forms defining hyperplanes
Outputs:
- a hyperplane arrangement, the hyperplane arrangement determined by L and R

Description

A hyperplane is a linear subspace of codimension one. An arrangement is a finite set of hyperplanes.

Probably the best-known hyperplane arrangement is the braid arrangement consisting of all the diagonal hyperplanes. In 4-space, it is constructed as follows:

i1 : S = ZZ[w,x,y,z];

i2 : A3 = arrangement {w-x,w-y,w-z,x-y,x-z,y-z}

o2 = {w - x, w - y, w - z, x - y, x - z, y - z}

o2 : Hyperplane Arrangement

i3 : describe A3

o3 = {w - x, w - y, w - z, x - y, x - z, y - z}

If we project along onto a subspace, then we obtain an essential arrangement:

i4 : R = S/ideal(w+x+y+z)

o4 = R

o4 : QuotientRing

i5 : A3' = arrangement({w-x,w-y,w-z,x-y,x-z,y-z},R)

o5 = {- 2x - y - z, - x - 2y - z, - x - y - 2z, x - y, x - z, y - z}

o5 : Hyperplane Arrangement

i6 : describe A3'

o6 = {- 2x - y - z, - x - 2y - z, - x - y - 2z, x - y, x - z, y - z}

The trivial arrangement has no equations.

i7 : trivial = arrangement({},S)

o7 = {}

o7 : Hyperplane Arrangement

i8 : describe trivial

o8 = {}

i9 : ring trivial

o9 = S

o9 : PolynomialRing

Caveat

If the elements of L are not ring elements in R, then the induced identity map is used to map them from ring L#0 into R.

Ways to use `arrangement` :

arrangement(Arrangement,Ring)
arrangement(List)
arrangement(List,Ring)
arrangement(Matrix)
arrangement(Matrix,Ring)
arrangement(Flat) (missing documentation)
arrangement(String) (missing documentation)
arrangement(String,PolynomialRing) -- look up a built-in hyperplane arrangement
arrangement(String,Ring) (missing documentation)

arrangement -- create a hyperplane arrangement

Synopsis

Description

Caveat

See also

Ways to use arrangement :

Ways to use `arrangement` :